Concept explainers
Review. As a sound wave passes through a gas, the compressions are either so rapid or so far apart that thermal
where M is the molar mass. The speed of sound in a gas is given by Equation 16.35; use that equation and the definition of the bulk modulus from Section 12.4. (b) Compute the theoretical speed of sound in air at 20.0°C and state how it compares with the value in Table 16.1. Take M = 28.9 g/mol. (c) Show that the speed of sound in an ideal gas is
where m0 is the mass of one molecule. (d) Slate how the result in part (c) compares with the most probable, average, and rms molecular speeds.
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Chapter 20 Solutions
Physics for Scientists and Engineers
- Consider the Maxwell-Boltzmann distribution function plotted in Problem 28. For those parameters, determine the rms velocity and the most probable speed, as well as the values of f(v) for each of these values. Compare these values with the graph in Problem 28. 28. Plot the Maxwell-Boltzmann distribution function for a gas composed of nitrogen molecules (N2) at a temperature of 295 K. Identify the points on the curve that have a value of half the maximum value. Estimate these speeds, which represent the range of speeds most of the molecules are likely to have. The mass of a nitrogen molecule is 4.68 1026 kg. Equation 20.18 can be used to find the rms velocity given the temperature, Boltzmanns constant, and the mass of the atom or molecule. The mass of a nitrogen molecule is 4.68 1026 kg. vrms=3kBTm=3(1.381023J/K)4.681026kg=511m/s Using the results of Problem 28 and the rms velocity, we can calculate the value of f(v). f(vrms) = (3.11 108)(511)2 e(5.75106(511)2) = 0.00181 The most probable speed, for which this function has its maximum value, is given by Equation 20.20. vmp=2kBTm=2(1.381023J/K)(295K)4.681026kg=417m/s f(vmp) = (3.11108)(417)2 e(5.75106(417)2) = 0.00199 We plot these points on the speed distribution. The most probable speed is indeed at the peak of the distribution function. Since the function is not symmetric, the rms velocity is somewhat higher than the most probable speed. Figure P20.29ANSarrow_forward2.00-mol of a monatomic ideal gas goes from State A to State D via the path A→B→C→D: State A PA=13.0atm, VA=13.00L State B PB=13.0atm, VB=6.00L State C PC=24.5atm, VC=6.00L State D PD=24.5atm, VD=21.50L Assume that the external pressure is constant during each step and equals the final pressure of the gas for that step. Calculate q for this process. Calculate w for this process .Calculate ΔE for this process Calculate ΔH for this process.arrow_forwardA sealed container contains a fixed volume of a monatomic ideal gas. If the gas temperature is increased by a factor of two, what is the ratio of the final to the initial (a) pressure, (b) average molecular kinetic energy, (c) root-mean-square speed, and (d) internal energy.arrow_forward
- 2.00-mol of a monatomic ideal gas goes from State A to State D via the path A→B→C→D: State A PA=11.5atm, VA=11.00L State B PB=11.5atm, VB=6.50L State C PC=20.5atm, VC=6.50L State D PD=20.5atm, VD=22.00L Assume that the external pressure is constant during each step and equals the final pressure of the gas for that step. Calculate ΔH for this process.arrow_forward2.00-mol of a monatomic ideal gas goes from State A to State D via the path A→B→C→D: State A PA=10.5atm, VA=13.50L State B PB=10.5atm, VB=6.00L State C PC=20.5atm, VC=6.00L State D PD=20.5atm, VD=25.00L Assume that the external pressure is constant during each step and equals the final pressure of the gas for that step. A) Calculate q for this process B) Calculate w for this process C) Calculate ΔE for this process D) Calculate ΔH for this process Express answer in L atm units!arrow_forward2.00-mol of a monatomic ideal gas goes from State A to State D via the path A→B→C→D: State A PA=10.5atm, VA=13.50L State B PB=10.5atm, VB=6.00L State C PC=20.5atm, VC=6.00L State D PD=20.5atm, VD=25.00L Assume that the external pressure is constant during each step and equals the final pressure of the gas for that step. A) Calculate q for this process B) Calculate w for this process C) Calculate ΔE for this process D) Calculate ΔH for this processarrow_forward
- Only answer letters (d) and (e) a) Using the Maxwell-Boltzmann function, calculate the fraction of argon gas molecules with a speed of 305 m/s at 500 K. (ans: 0.0014) b) If the system in (a) has 0.46 moles of argon gas, how many molecules have the speed of 305 m/s? (ans: 6.44 x 10-4) c) Calculate the values of vmp, vavg, and vrms for xenon gas at 298 K. (ans: Vmp=194.2 m/s , Vavg=219.2 m/s , Vrms=237.8 m/s) d) From the values calculated in (c), label the Boltzmann distribution plot with the approximate locations of vmp, v avg, and vrms e) What will have a larger speed distribution, helium at 500 K or argon at 300 K? Helium at 300 K or argon at 500 K? Argon at 400 K or argon at 1000 K?arrow_forwardA sealed cubical container 19.0 cm on a side contains a gas with five times Avogadro's number of krypton atoms at a temperature of 15.0°C. HINT (a) Find the internal energy (in J) of the gas. J (b) The total translational kinetic energy (in J) of the gas. J (c) Calculate the average kinetic energy (in J) per atom. J (d) Use P = 2 3 N V 1 2 mv2 to calculate the gas pressure (in Pa). Pa (e) Calculate the gas pressure (in Pa) using the ideal gas law (PV = nRT). Paarrow_forwardSuppose that a gas obeys at low pressures the equation PVm = RT + (A+ B/T2)P where A and B are constants independent of pressure and temperature, and Vm is the molar volume. Derive the expression for the change in enthalpy which will accompany the expansion of n moles of gas from a pressure P1to a pressure P2 at temperature T.arrow_forward
- A sealed cubical container 11.0 cm on a side contains a gas with five times Avogadro's number of neon atoms at a temperature of 25.0°C. (a) Find the internal energy (in J) of the gas. J (b) The total translational kinetic energy (in J) of the gas. J (c)Calculate the average kinetic energy (in J) per atom.J (d)Use P = 2 3 N V 1 2 mv2 to calculate the gas pressure (in Pa). (e)Calculate the gas pressure (in Pa) using the ideal gas law (PV = nRT). Paarrow_forwardFind the molar mass and the number of degrees of freedom of molecules in a gas if cV=0.65J/gK and cP=0.91J/g K.arrow_forwardA sealed cubical container 13.0 cm on a side contains a gas with four times Avogadro's number of xenon atoms at a temperature of 28.0°C. HINT (a) Find the internal energy (in J) of the gas. J (b) The total translational kinetic energy (in J) of the gas. J (c) Calculate the average kinetic energy (in J) per atom. J (d) Use P = 2 3 N V 1 2 mv2 to calculate the gas pressure (in Pa). Pa (e) Calculate the gas pressure (in Pa) using the ideal gas law (PV = nRT).arrow_forward
- Physics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage Learning